Probability. Engr. Jeffrey T. Dellosa.


 Reynard Strickland
 3 years ago
 Views:
Transcription
1 Probability Engr. Jeffrey T. Dellosa
2 Outline Probability 2.1 Sample Space 2.2 Events 2.3 Counting Sample Points 2.4 Probability of an Event 2.5 Additive Rules 2.6 Conditional Probability, Independence, and the Product Rule 2.7 Bayes Rule
3 SAMPLE SPACE Definition 2.1: The set of all possible outcomes of a statistical experiment is called the sample space and is represented by the symbol S. Each outcome in a sample space is called an element or a member of the sample space, or simply a sample point.
4 SAMPLE SPACE Thus, the sample space S, of possible outcomes when a coin is flipped, may be written S = {H, T} where H and T correspond to heads and tails, respectively.
5 Example 2.1 Consider the experiment of tossing a die. If we are interested in the number that shows on the top face, the sample space is S1 = {1, 2, 3, 4, 5, 6}. If we are interested only in whether the number is even or odd, the sample space is simply S2 = {even, odd}.
6 Example 2.2 An experiment consists of flipping a coin and then flipping it a second time if a head occurs. If a tail occurs on the first flip, then a die is tossed once. To list the elements of the sample space providing the most information, we construct the tree diagram of Figure 2.1.
7 Example 2.2
8 Example 2.2 By proceeding along all paths, we see that the sample space is S = {HH, HT, T1, T2, T3, T4, T5, T6}.
9 Example 2.2 Suppose that three items are selected at random from a manufacturing process. Each item is inspected and classified defective, D, or non defective, N. To list the elements of the sample space providing the most information, we construct the tree diagram of Figure 2.2.
10 Example 2.2
11 Example 2.2 we see that the sample space is S = {DDD, DDN, DND, DNN, NDD, NDN, NND, NNN}
12 Example 2.3 Sample spaces with a large or infinite number of sample points are best described by a statement or rule method. For example, if the possible outcomes of an experiment are the set of cities in the world with a population over 1 million, our sample space is written S = {x x is a city with a population over 1 million}, which reads S is the set of all x such that x is a city with a population over 1 million.
13 Definition 2.2 An event is a subset of a sample space.
14 Definition 2.2 Example 2.4: Given the sample space S = {t t 0}, where t is the life in years of a certain electronic component, then the event A that the component fails before the end of the fifth year is the subset A = {t 0 t < 5}.
15 Definition 2.2 It is conceivable that an event may be a subset that includes the entire sample space S or a subset of S called the null set and denoted by the symbol φ, which contains no elements at all. For instance, if we let A be the event of detecting a microscopic organism by the naked eye in a biological experiment, then A = φ. Also, if B = {x x is an even factor of 7}, then B must be the null set, since the only possible factors of 7 are the odd numbers 1 and 7.
16 Definition 2.2 Consider an experiment where the smoking habits of the employees of a manufacturing firm are recorded. A possible sample space might classify an individual as a nonsmoker, a light smoker, a moderate smoker, or a heavy smoker. Let the subset of smokers be some event. Then all the nonsmokers correspond to a different event, also a subset of S, which is called the complement of the set of smokers.
17 Definition 2.3 The complement of an event A with respect to S is the subset of all elements of S that are not in A. We denote the complement of A by the symbol A.
18 Definition 2.3 Example 2.5: Let R be the event that a red card is selected from an ordinary deck of 52 playing cards, and let S be the entire deck. Then R is the event that the card selected from the deck is not a red card but a black card.
19 Definition 2.3 Example 2.6: Consider the sample space S = {book, cell phone, mp3, paper, stationery, laptop}. Let A = {book, stationery, laptop, paper}. Then the complement of A is A = {cell phone, mp3}.
20 Definition 2.4 Definition 2.4: The intersection of two events A and B, denoted by the symbol A B, is the event containing all elements that are common to A and B.
21 Definition 2.4 Example 2.7: Let E be the event that a person selected at random in a classroom is majoring in engineering, and let F be the event that the person is female. Then E F is the event of all female engineering students in the classroom.
22 Definition 2.4 Example 2.8: Let V = {a, e, i, o, u} and C = {l, r, s, t}; then it follows that V C = φ. That is, V and C have no elements in common and, therefore, cannot both simultaneously occur.
23 Definition 2.5 Definition 2.5: Two events A and B are mutually exclusive, or disjoint, if A B = φ, that is, if A and B have no elements in common.
24 Definition 2.5 Example 2.9: A cable television company offers programs on eight different channels, three of which are affiliated with ABC, two with NBC, and one with CBS. The other two are an educational channel and the ESPN sports channel. Suppose that a person subscribing to this service turns on a television set without first selecting the channel. Let: A be the event that the program belongs to the NBC network and B the event that it belongs to the CBS network. Since a television program cannot belong to more than one network, the events A and B have no programs in common. Therefore, the intersection A B contains no programs, and consequently the events A and B are mutually exclusive. Often one is interested in the occurrence of at least one of two events associated with an experiment.
25 Definition 2.5 Thus, in the dietossing experiment: if A = {2, 4, 6} and B = {4, 5, 6}, we might be interested in either A or B occurring or both A and B occurring. Such an event, called the union of A and B, will occur if the outcome is an element of the subset {2, 4, 5, 6}.
26 Definition 2.6 Definition 2.6: The union of the two events A and B, denoted by the symbol A B, is the event containing all the elements that belong to A or B or both.
27 Definition 2.6 Example 2.10: Let A = {a, b, c} and B = {b, c, d, e}; then A B = {a, b, c, d, e}.
28 Definition 2.6 Example 2.12: If M = {x 3 < x < 9} and N = {y 5 < y < 12}, then M N = {z 3 < z < 12}.
29 Definition 2.6 The relationship between events and the corresponding sample space can be illustrated graphically by means of Venn diagrams. In a Venn diagram we let the sample space be a rectangle and represent events by circles drawn inside the rectangle.
30 Definition 2.6
31 Definition 2.6 Thus, in Figure 2.3, we see that: A B = regions 1 and 2, B C = regions 1 and 3,
32 Definition 2.6 Thus, in Figure 2.3, we see that: A C = regions 1, 2, 3, 4, 5, and 7, B A = regions 4 and 7,
33 Definition 2.6 Thus, in Figure 2.3, we see that: A B C = region 1, (A B) C = regions 2, 6, and 7,
34 Definition 2.6 In Figure 2.4, we see that events A, B, and C are all subsets of the sample space S. It is also clear that event B is a subset of event A; event B C has no elements and hence B and C are mutually exclusive; Event A C has at least one element; and Event A B = A.
35 Definition 2.6 Figure 2.4 might, therefore, depict a situation where we select a card at random from an ordinary deck of 52 playing cards and observe whether the following events occur: A: the card is red, B: the card is the jack, queen, or king of diamonds, C: the card is an ace. Clearly, the event A C consists of only the two red aces.
36 ASSIGNMENT 2.1 List the elements of each of the following sample spaces: (a) the set of integers between 1 and 50 divisible by 8; (b) the set S = {x x2 + 4x 5 = 0}; (c) the set of outcomes when a coin is tossed until a tail or three heads appear; (d) the set S = {x x is a continent}; (e) the set S = {x 2x 4 0 and x < 1}.
37 ASSIGNMENT 2.3 Which of the following events are equal? (a) A = {1, 3}; (b) B = {x x is a number on a die}; (c) C = {x x2 4x+3 = 0}; (d) D = {x x is the number of heads when six coins are tossed}.
38 2.3 Counting Sample Points RULE 2.1 Rule 2.1: If an operation can be performed in n 1 ways, and if for each of these ways a second operation can be performed in n 2 ways, then the two operations can be performed together in n 1 n 2 ways.
39 2.3 Counting Sample Points Example 2.13: How many sample points are there in the sample space when a pair of dice is thrown once? Solution : The first die can land faceup in any one of n1 = 6 ways. For each of these 6 ways, the second die can also land faceup in n2 = 6 ways. Therefore, the pair of dice can land in n1n2 = (6)(6) = 36 possible ways.
40 2.3 Counting Sample Points Example 2.14: A developer of a new subdivision offers prospective home buyers a choice of Tudor, rustic, colonial, and traditional exterior styling in ranch, twostory, and splitlevel floor plans. In how many different ways can a buyer order one of these homes?
41 Solution : Since n1 = 4 and n2 = 3, a buyer must choose from n1n2 = (4)(3) = 12 possible homes.
42 2.3 Counting Sample Points RULE 2.2 Rule 2.2: If an operation can be performed in n1 ways, and if for each of these a second operation can be performed in n2 ways, and for each of the first two a third operation can be performed in n3 ways, and so forth, then the sequence of k operations can be performed in n1n2 nk ways.
43 2.3 Counting Sample Points RULE 2.2 Example 2.16: Sam is going to assemble a computer by himself. He has the choice of: chips from two (2) brands, a hard drive from four (4), memory from three (3), and an accessory bundle from five (5) local stores. How many different ways can Sam order the parts? Solution : Since n1 = 2, n2 = 4, n3 = 3, and n4 = 5, there are nl n2 n3 n4 = = 120 different ways to order the parts.
44 2.3 Counting Sample Points Definition 2.7: A permutation is an arrangement of all or part of a set of objects.
45 2.3 Counting Sample Points PERMUTATION Consider the three letters a, b, and c. The possible permutations are abc, acb, bac, bca, cab, and cba. Thus, we see that there are 6 distinct arrangements. No matter which two letters are chosen for the first two positions, there is only n3 = 1 choice for the last position, giving a total of n1n2n3 = (3)(2)(1) = 6 permutations
46 2.3 Counting Sample Points PERMUTATION Definition 2.8: For any nonnegative integer n, n!, called n factorial, is defined as: n! = n(n 1) (2)(1), with special case 0! = 1. Theorem 2.1: The number of permutations of n objects is n!.
47 2.3 Counting Sample Points PERMUTATION The number of permutations of the four letters a, b, c, and d will be 4! = 24. Now consider the number of permutations that are possible by taking two letters at a time from four. These would be ab, ac, ad, ba, bc, bd, ca, cb, cd, da, db, and dc. Using Rule 2.1 again, we have two positions to fill, with n1 = 4 choices for the first and then n2 = 3 choices for the second, for a total of: n1n2 = (4)(3) = 12 permutations
48 2.3 Counting Sample Points PERMUTATION In general, n distinct objects taken r at a time can be arranged in: n(n 1)(n 2) (n r + 1) ways. We represent this product by the symbol:
49 2.3 Counting Sample Points PERMUTATION Example 2.18: In one year, three awards (research, teaching, and service) will be given to a class of 25 graduate students in a statistics department. If each student can receive at most one award, how many possible selections are there? Solution : Since the awards are distinguishable, it is a permutation problem. The total number of sample points is:
50 PROBABILITY This is an alternate lecture
51 1. Introduction Probability theory is devoted to the study of uncertainty and variability Tasks: 1. Basic of probability: rules, terminology, the basic calculus of probability 2. Random variables, expectations, variances 3. Simulation
52 Main Concepts 1) Probability in this course represents the relative frequency of outcomes after a great many (infinity many) repetitions. 2) We study the probability because it is a tool that let us make an inference from a sample to a population 3) Probability is used to understand what patterns in nature are real and which are due to chance 4) Independent is the fundamental concept of probability & statistics 5) Conditional probability is also fundamental importance in part because it help us understand independence
53 Sample Space Probability: quantify the variability in the outcome of any experiment whose exact outcome cannot be predicted with certainty. The Space of outcome!! Sample space: a set of all possible outcomes of an experiment. Usually denoted by S.
54 Example Throw a coin S = {H, T} Throw a coin twice S={HH, HT, TH, TT} 7 race horses {1, 2, 3, 4, 5, 6,7} S={all 7! Permutation of {1,...,7}
55 Sample Space Finite sample space: finite number of elements in the space S. Countable infinite sample space: ex. natural numbers. Discrete sample space: if it has finite many or countable infinity of elements. Continuous sample space: If the elements constitute a continuum. Ex. All the points in a line.
56 Event Event: subset of a sample space. In words, an event A is a set (or group) of possible outcomes of an uncertain process e.g. {Heads in a single coin toss}, {rain}. Example: A government agency must decide where to locate two new computer research facilities (Vancouver, Toronto). C={(1,0), (0,1)} is the event that between them, Vancouver and Toronto will get one. S={(0,0), (0,1), (0,2),(1,0),(1,1),(2,0)}
57 Mutually exclusive events Mutually exclusive: Two events have no elements in common. Ex. C={(1,0), (0,1)}, D={(0,0),(0,1), (0,2)}, E={(0,0), (1,1)} Then C and E are, while D and E are not.
58 Events Union: A B subset of S that contains all elements that are either in A or B, or both. Intersection: A B subset of S that contains all elements that are in both A and B Complement: A subset of S that contains elements that are not in A
59 Venn Diagrams Set A and B A B Set A or B A B
60 Venn Diagrams Not A A
61 DeMorgan s Law n n ( i 1 E i ) i 1 E i n ( E ) i 1 i n i 1 E i
62 What is Probability defined? Classical probability concept Frequency interpretation Subjective probability Axiom of probability
63 Probability Classical probability concept: If there are n equally likely possibilities, of which one must occur, and x are regarded as favorable, or as a success, the probability of a success is given by x/ n. Ex. The probability of drawing an ace from a well shuffled deck of 52 playing cards. 4/52.
64 Limitation Limited of classical probability: many situations in which the various possibility cannot be regarded as equally likely. Ex. Election.
65 Frequency interpretation The probability of an event (or outcome) is the proportion of times the event occur in a long run of repeated experiment.
66 Subjective probability Probabilities: personal or subjective evaluations. Express the strength of one s belief with regard to the uncertainties that are involved.
67 Axiom of Probability Axiom 1. in S. 0 P( A) 1 for each event A Axiom 2. P(S) = 1 Axiom 3. If A and B are mutually exclusive events in S, then P( A B) P( A) P( B)
68 Checking probabilities Example: P69, An experiment has the three possible and mutually exclusive outcomes A, B, C. Check the assignment of probabilities is permissible: P( A), P( B), P( C) P( A) 0.57, P( B) 0.24, P( C) 0.19
69 Counting  Combinatorial analysis Goal: Determine the number of elements in a finite sample space (or in a event). Example: P58. A consumer testing service rates lawn mowers: 1) operate: easy, average, difficult 2) price: expensive, inexpensive 3) repair: costly, average, cheap Q: How many different ways can a law mower be rated by this testing service?
70 Tree diagram O1 P1 P2 r1 r2 r3 r1 r2 r3 O2 O3 P1 P2 P1 P2 r1 r2 r3 r1 r2 r3 r1 r2 r3 r1 r2 r3
71 Tree diagram A given path from left to right along the branches of the trees, we obtain an element of the sample space Total number of path in a tree is equal to the total number of elements in the sample space.
72 Multiplication of choice Theorem 3.1. If sets A 1, A 2,..., A k contain, respectively, n 1, n 2,..., n k elements, there are n 1 n 2...n k ways of choosing first an element of A 1, then an element of A 2,..., and finally an element of A k.
73 Permutation Permutation: r objects are chosen from a set of n distinct objects, any particular arrangement, or order of these objects. Total number of permutation r from n objects. np r n( n 1)( n 2)...( n r 1)
74 Factorial notation 1! = 1, 2! = 2*1 =2, 3!=3*2*1=6. n! n( n 1)...1 Let 0!=1. np r ( n n! r)!
75 Combinations Combinations of n objects taken r at a time. n r npr n! r! ( n r)! r! r objects from n, but don t care about the order of these r objects.
76 EX. contrast 12 12P2 12*11 132, C2 12*11/ 2 66 Please calculate: 12P P P,12, How fast factorial grow and the impact that considering order has.
77 Examples Urn Problem : Suppose we have an urn with 30 blue balls and 50 red balls in it and that these balls are identical except for color. Suppose further the balls are well mixed and that we draw 3 balls, without replacement. Determine the probability that the balls are all of the same color.
78 Element Theorem Theorem 3.4. If A, A, 1 2, A n are mutually exclusive events in a sample space S, then P( A A2 An ) P( A1 ) P( A 1 n ) Proposition 1. P( A) 1 P( A)
79 Propositions Proposition 2. If A B, then P( A) P( B) Proposition 3. If A and B are any events in S, then P( A B) P( A) P( B) P( A B) Proof Sketch: 1. Apply the formula of exercise 3.13 (c) and (d) 2. Apply theorem 3.4
80 Example: Suppose that we toss two coins and suppose that each of the four points in the sample space S={(H,H), (H, T), (T, H), (T, T)} is equally likely and hence has probability ¼. Let E = {(H, H), (H, T)} and F ={(H, H), (T, H)}. What is the probability of P(E or F)?
81 Extension Discussion: P( A B C)? P( A B C) P( A) P( B) P( C) P( AB) P( AC) P( BC) P( ABC) P( A B C D )??? n r 1 ( 1 2 n) ( i) ( i ) ( 1) ( ) 1 i 2 i1 i2 ir i 1 i i i i i P E E E P E P E E P E E E n 1 ( 1) P( E1E 2 En) r
82 Counting  continuous Binomial theorem n n n k n k ( x y) x y k 0 k Multinomial coefficient A set of n distinct item is to be divided into r distinct groups of respective n1 n2 r sizes, where i 1 n i n How many different divisions are possible?,,, nr
83 Multinomial coefficients n n! n, n,, n!!! 1 2 r n n n 1 2 r n ( ) 1 2 r n 1 2 x x x x x x n n n 1 2 r 1 2 r n n n n n1, n2,, nr r
84 Examples In the game of bridge the entire deck of 52 cards is dealt out to 4 players. What is the probability that (a) one of the player receives all 13 spades; (b) each player receives 1 ace?
85 Solution (a) ,13,13 6.3* ,13,13,13 12 (b) 48 4! 12,12,12, ,13,13,13
86 EX. In the game of bridge the entire deck of 52 cards is dealt out to 4 players. What is the probability that the diamonds in 4 players are ?
87 Examples A poker hand consists of 5 cards. What is the probability that one is dealt a straight? 5 10(4 4)
88 Examples What is the probability that one is dealt a full house?
89 Ex. If n people are presented in a room, what is probability that no two of them celebrate their birthday on the same day of the year? How large need n be so that this probability is less than ½? n 23
90 Probability and a paradox Suppose we posses an infinite large urn and an infinite collection of balls labeled ball number 1, number 2, and so on. Experiment: At 1 minute to 12P.M., ball numbered 1 through 10 are placed in the urn, and ball number 10 is withdrawn. At ½ minute to 12 P.M., ball numbered 11 through 20 are placed in the urn, and ball number 20 is withdraw. At ¼ minute to 12 P.M., and so on. Question: how many balls are in the urn at 12 P.M.?
91 Paradox Empty Any number n is withdraw in Another (1/2) (n1). experiment: The balls are withdraw begins from 1, n 2 Infinite of course! Define E to be the event that ball number What is case that arbitrarily 1 is still choose the withdraw? in the urn after the first n withdrawals have been made. PE ( ) n (9 n) (9n 1)
92 Case study Randomized quick sort algorithm Expected number of comparisons
93 Summary Sample space specifies all possible outcomes. Always assign probabilities to events that satisfy the axioms of probability.
94 Homework Problems in Textbook (3.7,3.16,3.31,3.34,3.37)
Chapter 1. Probability
Chapter 1. Probability 1.1 Basic Concepts Scientific method a. For a given problem, we define measures that explains the problem well. b. Data is collected with observation and the measures are calculated.
More informationChapter 1. Probability
Chapter 1. Probability 1.1 Basic Concepts Scientific method a. For a given problem, we define measures that explains the problem well. b. Data is collected with observation and the measures are calculated.
More informationCHAPTER 2 PROBABILITY. 2.1 Sample Space. 2.2 Events
CHAPTER 2 PROBABILITY 2.1 Sample Space A probability model consists of the sample space and the way to assign probabilities. Sample space & sample point The sample space S, is the set of all possible outcomes
More informationContents 2.1 Basic Concepts of Probability Methods of Assigning Probabilities Principle of Counting  Permutation and Combination 39
CHAPTER 2 PROBABILITY Contents 2.1 Basic Concepts of Probability 38 2.2 Probability of an Event 39 2.3 Methods of Assigning Probabilities 39 2.4 Principle of Counting  Permutation and Combination 39 2.5
More informationWeek 3 Classical Probability, Part I
Week 3 Classical Probability, Part I Week 3 Objectives Proper understanding of common statistical practices such as confidence intervals and hypothesis testing requires some familiarity with probability
More informationBusiness Statistics. Chapter 4 Using Probability and Probability Distributions QMIS 120. Dr. Mohammad Zainal
Department of Quantitative Methods & Information Systems Business Statistics Chapter 4 Using Probability and Probability Distributions QMIS 120 Dr. Mohammad Zainal Chapter Goals After completing this chapter,
More informationThe probability setup
CHAPTER 2 The probability setup 2.1. Introduction and basic theory We will have a sample space, denoted S (sometimes Ω) that consists of all possible outcomes. For example, if we roll two dice, the sample
More informationElementary Combinatorics
184 DISCRETE MATHEMATICAL STRUCTURES 7 Elementary Combinatorics 7.1 INTRODUCTION Combinatorics deals with counting and enumeration of specified objects, patterns or designs. Techniques of counting are
More informationThe probability setup
CHAPTER The probability setup.1. Introduction and basic theory We will have a sample space, denoted S sometimes Ω that consists of all possible outcomes. For example, if we roll two dice, the sample space
More informationChapter 5  Elementary Probability Theory
Chapter 5  Elementary Probability Theory Historical Background Much of the early work in probability concerned games and gambling. One of the first to apply probability to matters other than gambling
More informationECON 214 Elements of Statistics for Economists
ECON 214 Elements of Statistics for Economists Session 4 Probability Lecturer: Dr. Bernardin Senadza, Dept. of Economics Contact Information: bsenadza@ug.edu.gh College of Education School of Continuing
More informationNovember 6, Chapter 8: Probability: The Mathematics of Chance
Chapter 8: Probability: The Mathematics of Chance November 6, 2013 Last Time Crystallographic notation Groups Crystallographic notation The first symbol is always a p, which indicates that the pattern
More informationTheory of Probability  Brett Bernstein
Theory of Probability  Brett Bernstein Lecture 3 Finishing Basic Probability Review Exercises 1. Model flipping two fair coins using a sample space and a probability measure. Compute the probability of
More informationChapter 3: Elements of Chance: Probability Methods
Chapter 3: Elements of Chance: Methods Department of Mathematics Izmir University of Economics Week 34 20142015 Introduction In this chapter we will focus on the definitions of random experiment, outcome,
More informationRANDOM EXPERIMENTS AND EVENTS
Random Experiments and Events 18 RANDOM EXPERIMENTS AND EVENTS In daytoday life we see that before commencement of a cricket match two captains go for a toss. Tossing of a coin is an activity and getting
More informationIntermediate Math Circles November 1, 2017 Probability I
Intermediate Math Circles November 1, 2017 Probability I Probability is the study of uncertain events or outcomes. Games of chance that involve rolling dice or dealing cards are one obvious area of application.
More informationMath 227 Elementary Statistics. Bluman 5 th edition
Math 227 Elementary Statistics Bluman 5 th edition CHAPTER 4 Probability and Counting Rules 2 Objectives Determine sample spaces and find the probability of an event using classical probability or empirical
More informationCHAPTER 8 Additional Probability Topics
CHAPTER 8 Additional Probability Topics 8.1. Conditional Probability Conditional probability arises in probability experiments when the person performing the experiment is given some extra information
More informationProbability (Devore Chapter Two)
Probability (Devore Chapter Two) 101635101 Probability Winter 20112012 Contents 1 Axiomatic Probability 2 1.1 Outcomes and Events............................... 2 1.2 Rules of Probability................................
More informationI. WHAT IS PROBABILITY?
C HAPTER 3 PROAILITY Random Experiments I. WHAT IS PROAILITY? The weatherman on 10 o clock news program states that there is a 20% chance that it will snow tomorrow, a 65% chance that it will rain and
More informationPROBABILITY. 1. Introduction. Candidates should able to:
PROBABILITY Candidates should able to: evaluate probabilities in simple cases by means of enumeration of equiprobable elementary events (e.g for the total score when two fair dice are thrown), or by calculation
More informationWeek 1: Probability models and counting
Week 1: Probability models and counting Part 1: Probability model Probability theory is the mathematical toolbox to describe phenomena or experiments where randomness occur. To have a probability model
More informationTextbook: pp Chapter 2: Probability Concepts and Applications
1 Textbook: pp. 3980 Chapter 2: Probability Concepts and Applications 2 Learning Objectives After completing this chapter, students will be able to: Understand the basic foundations of probability analysis.
More informationProbability. Ms. Weinstein Probability & Statistics
Probability Ms. Weinstein Probability & Statistics Definitions Sample Space The sample space, S, of a random phenomenon is the set of all possible outcomes. Event An event is a set of outcomes of a random
More informationSection Introduction to Sets
Section 1.1  Introduction to Sets Definition: A set is a welldefined collection of objects usually denoted by uppercase letters. Definition: The elements, or members, of a set are denoted by lowercase
More informationProbability and Counting Techniques
Probability and Counting Techniques Diana Pell (Multiplication Principle) Suppose that a task consists of t choices performed consecutively. Suppose that choice 1 can be performed in m 1 ways; for each
More informationBasic Probability Models. PingShou Zhong
asic Probability Models PingShou Zhong 1 Deterministic model n experiment that results in the same outcome for a given set of conditions Examples: law of gravity 2 Probabilistic model The outcome of the
More informationDefine and Diagram Outcomes (Subsets) of the Sample Space (Universal Set)
12.3 and 12.4 Notes Geometry 1 Diagramming the Sample Space using Venn Diagrams A sample space represents all things that could occur for a given event. In set theory language this would be known as the
More information3 The multiplication rule/miscellaneous counting problems
Practice for Exam 1 1 Axioms of probability, disjoint and independent events 1 Suppose P (A 0, P (B 05 (a If A and B are independent, what is P (A B? What is P (A B? (b If A and B are disjoint, what is
More informationIf you roll a die, what is the probability you get a four OR a five? What is the General Education Statistics
If you roll a die, what is the probability you get a four OR a five? What is the General Education Statistics probability that you get neither? Class Notes The Addition Rule (for OR events) and Complements
More information3 The multiplication rule/miscellaneous counting problems
Practice for Exam 1 1 Axioms of probability, disjoint and independent events 1. Suppose P (A) = 0.4, P (B) = 0.5. (a) If A and B are independent, what is P (A B)? What is P (A B)? (b) If A and B are disjoint,
More informationProbability and Statistics. Copyright Cengage Learning. All rights reserved.
Probability and Statistics Copyright Cengage Learning. All rights reserved. 14.2 Probability Copyright Cengage Learning. All rights reserved. Objectives What Is Probability? Calculating Probability by
More informationINDIAN STATISTICAL INSTITUTE
INDIAN STATISTICAL INSTITUTE B1/BVR Probability Home Assignment 1 200707 1. A poker hand means a set of five cards selected at random from usual deck of playing cards. (a) Find the probability that it
More informationCombinatorics and Intuitive Probability
Chapter Combinatorics and Intuitive Probability The simplest probabilistic scenario is perhaps one where the set of possible outcomes is finite and these outcomes are all equally likely. A subset of the
More informationSection : Combinations and Permutations
Section 11.111.2: Combinations and Permutations Diana Pell A construction crew has three members. A team of two must be chosen for a particular job. In how many ways can the team be chosen? How many words
More informationImportant Distributions 7/17/2006
Important Distributions 7/17/2006 Discrete Uniform Distribution All outcomes of an experiment are equally likely. If X is a random variable which represents the outcome of an experiment of this type, then
More informationProbability. Dr. Zhang Fordham Univ.
Probability! Dr. Zhang Fordham Univ. 1 Probability: outline Introduction! Experiment, event, sample space! Probability of events! Calculate Probability! Through counting! Sum rule and general sum rule!
More informationCompound Probability. Set Theory. Basic Definitions
Compound Probability Set Theory A probability measure P is a function that maps subsets of the state space Ω to numbers in the interval [0, 1]. In order to study these functions, we need to know some basic
More informationCSC/MATA67 Tutorial, Week 12
CSC/MATA67 Tutorial, Week 12 November 23, 2017 1 More counting problems A class consists of 15 students of whom 5 are prefects. Q: How many committees of 8 can be formed if each consists of a) exactly
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Study Guide for Test III (MATH 1630) Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the number of subsets of the set. 1) {x x is an even
More informationStatistics Intermediate Probability
Session 6 oscardavid.barrerarodriguez@sciencespo.fr April 3, 2018 and Sampling from a Population Outline 1 The Monty Hall Paradox Some Concepts: Event Algebra Axioms and Things About that are True Counting
More informationSuch a description is the basis for a probability model. Here is the basic vocabulary we use.
5.2.1 Probability Models When we toss a coin, we can t know the outcome in advance. What do we know? We are willing to say that the outcome will be either heads or tails. We believe that each of these
More informationIntroduction to probability
Introduction to probability Suppose an experiment has a finite set X = {x 1,x 2,...,x n } of n possible outcomes. Each time the experiment is performed exactly one on the n outcomes happens. Assign each
More information2. Combinatorics: the systematic study of counting. The Basic Principle of Counting (BPC)
2. Combinatorics: the systematic study of counting The Basic Principle of Counting (BPC) Suppose r experiments will be performed. The 1st has n 1 possible outcomes, for each of these outcomes there are
More informationBefore giving a formal definition of probability, we explain some terms related to probability.
probability 22 INTRODUCTION In our daytoday life, we come across statements such as: (i) It may rain today. (ii) Probably Rajesh will top his class. (iii) I doubt she will pass the test. (iv) It is unlikely
More informationClass XII Chapter 13 Probability Maths. Exercise 13.1
Exercise 13.1 Question 1: Given that E and F are events such that P(E) = 0.6, P(F) = 0.3 and P(E F) = 0.2, find P (E F) and P(F E). It is given that P(E) = 0.6, P(F) = 0.3, and P(E F) = 0.2 Question 2:
More informationChapter 1. Set Theory
Chapter 1 Set Theory 1 Section 1.1: Types of Sets and Set Notation Set: A collection or group of distinguishable objects. Ex. set of books, the letters of the alphabet, the set of whole numbers. You can
More informationProbability. The MEnTe Program Math Enrichment through Technology. Title V East Los Angeles College
Probability The MEnTe Program Math Enrichment through Technology Title V East Los Angeles College 2003 East Los Angeles College. All rights reserved. Topics Introduction Empirical Probability Theoretical
More informationProbability Theory. Mohamed I. Riffi. Islamic University of Gaza
Probability Theory Mohamed I. Riffi Islamic University of Gaza Table of contents 1. Chapter 1 Probability Properties of probability Counting techniques 1 Chapter 1 Probability Probability Theorem P(φ)
More informationWell, there are 6 possible pairs: AB, AC, AD, BC, BD, and CD. This is the binomial coefficient s job. The answer we want is abbreviated ( 4
2 More Counting 21 Unordered Sets In counting sequences, the ordering of the digits or letters mattered Another common situation is where the order does not matter, for example, if we want to choose a
More informationFinite Math  Fall 2016
Finite Math  Fall 206 Lecture Notes  /28/206 Section 7.4  Permutations and Combinations There are often situations in which we have to multiply many consecutive numbers together, for example, in examples
More informationThe study of probability is concerned with the likelihood of events occurring. Many situations can be analyzed using a simplified model of probability
The study of probability is concerned with the likelihood of events occurring Like combinatorics, the origins of probability theory can be traced back to the study of gambling games Still a popular branch
More informationProbability Models. Section 6.2
Probability Models Section 6.2 The Language of Probability What is random? Empirical means that it is based on observation rather than theorizing. Probability describes what happens in MANY trials. Example
More informationSTAT 3743: Probability and Statistics
STAT 3743: Probability and Statistics G. Jay Kerns, Youngstown State University Fall 2010 Probability Random experiment: outcome not known in advance Sample space: set of all possible outcomes (S) Probability
More informationCOUNTING AND PROBABILITY
CHAPTER 9 COUNTING AND PROBABILITY Copyright Cengage Learning. All rights reserved. SECTION 9.2 Possibility Trees and the Multiplication Rule Copyright Cengage Learning. All rights reserved. Possibility
More information, the of all of a probability experiment. consists of outcomes. (b) List the elements of the event consisting of a number that is greater than 4.
41 Sample Spaces and Probability as a general concept can be defined as the chance of an event occurring. In addition to being used in games of chance, probability is used in the fields of,, and forecasting,
More informationCS100: DISCRETE STRUCTURES. Lecture 8 Counting  CH6
CS100: DISCRETE STRUCTURES Lecture 8 Counting  CH6 Lecture Overview 2 6.1 The Basics of Counting: THE PRODUCT RULE THE SUM RULE THE SUBTRACTION RULE THE DIVISION RULE 6.2 The Pigeonhole Principle. 6.3
More informationChapter 2. Permutations and Combinations
2. Permutations and Combinations Chapter 2. Permutations and Combinations In this chapter, we define sets and count the objects in them. Example Let S be the set of students in this classroom today. Find
More informationSample Spaces, Events, Probability
Sample Spaces, Events, Probability CS 3130/ECE 3530: Probability and Statistics for Engineers August 28, 2014 Sets A set is a collection of unique objects. Sets A set is a collection of unique objects.
More informationChapter 4: Probability and Counting Rules
Chapter 4: Probability and Counting Rules Before we can move from descriptive statistics to inferential statistics, we need to have some understanding of probability: Ch4: Probability and Counting Rules
More informationLecture 6 Probability
Lecture 6 Probability Example: When you toss a coin, there are only two possible outcomes, heads and tails. What if we toss a coin two times? Figure below shows the results of tossing a coin 5000 times
More informationProbability MAT230. Fall Discrete Mathematics. MAT230 (Discrete Math) Probability Fall / 37
Probability MAT230 Discrete Mathematics Fall 2018 MAT230 (Discrete Math) Probability Fall 2018 1 / 37 Outline 1 Discrete Probability 2 Sum and Product Rules for Probability 3 Expected Value MAT230 (Discrete
More informationDiscrete Structures Lecture Permutations and Combinations
Introduction Good morning. Many counting problems can be solved by finding the number of ways to arrange a specified number of distinct elements of a set of a particular size, where the order of these
More informationCIS 2033 Lecture 6, Spring 2017
CIS 2033 Lecture 6, Spring 2017 Instructor: David Dobor February 2, 2017 In this lecture, we introduce the basic principle of counting, use it to count subsets, permutations, combinations, and partitions,
More informationThe next several lectures will be concerned with probability theory. We will aim to make sense of statements such as the following:
CS 70 Discrete Mathematics for CS Fall 2004 Rao Lecture 14 Introduction to Probability The next several lectures will be concerned with probability theory. We will aim to make sense of statements such
More information6/24/14. The Poker Manipulation. The Counting Principle. MAFS.912.SIC.1: Understand and evaluate random processes underlying statistical experiments
The Poker Manipulation Unit 5 Probability 6/24/14 Algebra 1 Ins1tute 1 6/24/14 Algebra 1 Ins1tute 2 MAFS. 7.SP.3: Investigate chance processes and develop, use, and evaluate probability models MAFS. 7.SP.3:
More informationProbability and Randomness. Day 1
Probability and Randomness Day 1 Randomness and Probability The mathematics of chance is called. The probability of any outcome of a chance process is a number between that describes the proportion of
More informationBlock 1  Sets and Basic Combinatorics. Main Topics in Block 1:
Block 1  Sets and Basic Combinatorics Main Topics in Block 1: A short revision of some set theory Sets and subsets. Venn diagrams to represent sets. Describing sets using rules of inclusion. Set operations.
More informationDiscrete Structures for Computer Science
Discrete Structures for Computer Science William Garrison bill@cs.pitt.edu 6311 Sennott Square Lecture #23: Discrete Probability Based on materials developed by Dr. Adam Lee The study of probability is
More informationCombinatorics: The Fine Art of Counting
Combinatorics: The Fine Art of Counting Week 6 Lecture Notes Discrete Probability Note Binomial coefficients are written horizontally. The symbol ~ is used to mean approximately equal. Introduction and
More informationChapter 2: Probability
Chapter 2: Probability Curtis Miller 20180613 Introduction Next we focus on probability. Probability is the mathematical study of randomness and uncertain outcomes. The subject may be as old as calculus.
More informationBIOL2300 Biostatistics Chapter 4 Counting and Probability
BIOL2300 Biostatistics Chapter 4 Counting and Probability Event, sample space sample space (generally denoted Ω, pronounced omega ): set of outcomes of a random experiment {H,T} set of coin flips {1,2,3,4,5,6}
More informationExercise Class XI Chapter 16 Probability Maths
Exercise 16.1 Question 1: Describe the sample space for the indicated experiment: A coin is tossed three times. A coin has two faces: head (H) and tail (T). When a coin is tossed three times, the total
More information4.1 Sample Spaces and Events
4.1 Sample Spaces and Events An experiment is an activity that has observable results. Examples: Tossing a coin, rolling dice, picking marbles out of a jar, etc. The result of an experiment is called an
More informationProbability  Chapter 4
Probability  Chapter 4 In this chapter, you will learn about probability its meaning, how it is computed, and how to evaluate it in terms of the likelihood of an event actually happening. A cynical person
More informationCSE 312: Foundations of Computing II Quiz Section #2: InclusionExclusion, Pigeonhole, Introduction to Probability (solutions)
CSE 31: Foundations of Computing II Quiz Section #: InclusionExclusion, Pigeonhole, Introduction to Probability (solutions) Review: Main Theorems and Concepts Binomial Theorem: x, y R, n N: (x + y) n
More informationChapter 5: Probability: What are the Chances? Section 5.2 Probability Rules
+ Chapter 5: Probability: What are the Chances? Section 5.2 + TwoWay Tables and Probability When finding probabilities involving two events, a twoway table can display the sample space in a way that
More informationIntroductory Probability
Introductory Probability Combinations Nicholas Nguyen nicholas.nguyen@uky.edu Department of Mathematics UK Agenda Assigning Objects to Identical Positions Denitions Committee Card Hands Coin Toss Counts
More informationChapter 6: Probability and Simulation. The study of randomness
Chapter 6: Probability and Simulation The study of randomness 6.1 Randomness Probability describes the pattern of chance outcomes. Probability is the basis of inference Meaning, the pattern of chance outcomes
More informationProbability and Counting Rules. Chapter 3
Probability and Counting Rules Chapter 3 Probability as a general concept can be defined as the chance of an event occurring. Many people are familiar with probability from observing or playing games of
More informationSTANDARD COMPETENCY : 1. To use the statistics rules, the rules of counting, and the characteristic of probability in problem solving.
Worksheet 4 th Topic : PROBABILITY TIME : 4 X 45 minutes STANDARD COMPETENCY : 1. To use the statistics rules, the rules of counting, and the characteristic of probability in problem solving. BASIC COMPETENCY:
More informationProblems from 9th edition of Probability and Statistical Inference by Hogg, Tanis and Zimmerman:
Math 22 Fall 2017 Homework 2 Drew Armstrong Problems from 9th edition of Probability and Statistical Inference by Hogg, Tanis and Zimmerman: Section 1.2, Exercises 5, 7, 13, 16. Section 1.3, Exercises,
More informationPROBABILITY. The sample space of the experiment of tossing two coins is given by
PROBABILITY Introduction Probability is defined as a quantitative measure of uncertainty a numerical value that conveys the strength of our belief in the occurrence of an event. The probability of an event
More informationThe Teachers Circle Mar. 20, 2012 HOW TO GAMBLE IF YOU MUST (I ll bet you $5 that if you give me $10, I ll give you $20.)
The Teachers Circle Mar. 2, 22 HOW TO GAMBLE IF YOU MUST (I ll bet you $ that if you give me $, I ll give you $2.) Instructor: Paul Zeitz (zeitzp@usfca.edu) Basic Laws and Definitions of Probability If
More information8.2 Union, Intersection, and Complement of Events; Odds
8.2 Union, Intersection, and Complement of Events; Odds Since we defined an event as a subset of a sample space it is natural to consider set operations like union, intersection or complement in the context
More informationCHAPTERS 14 & 15 PROBABILITY STAT 203
CHAPTERS 14 & 15 PROBABILITY STAT 203 Where this fits in 2 Up to now, we ve mostly discussed how to handle data (descriptive statistics) and how to collect data. Regression has been the only form of statistical
More informationCSE 21 Mathematics for Algorithm and System Analysis
CSE 21 Mathematics for Algorithm and System Analysis Unit 1: Basic Count and List Section 3: Set CSE21: Lecture 3 1 Reminder Piazza forum address: http://piazza.com/ucsd/summer2013/cse21/hom e Notes on
More informationIntroduction to Probability and Statistics I Lecture 7 and 8
Introduction to Probability and Statistics I Lecture 7 and 8 Basic Probability and Counting Methods Computing theoretical probabilities:counting methods Great for gambling! Fun to compute! If outcomes
More informationAxiomatic Probability
Axiomatic Probability The objective of probability is to assign to each event A a number P(A), called the probability of the event A, which will give a precise measure of the chance thtat A will occur.
More informationSTAT Statistics I Midterm Exam One. Good Luck!
STAT 515  Statistics I Midterm Exam One Name: Instruction: You can use a calculator that has no connection to the Internet. Books, notes, cellphones, and computers are NOT allowed in the test. There are
More informationProbability Concepts and Counting Rules
Probability Concepts and Counting Rules Chapter 4 McGrawHill/Irwin Dr. Ateq Ahmed AlGhamedi Department of Statistics P O Box 80203 King Abdulaziz University Jeddah 21589, Saudi Arabia ateq@kau.edu.sa
More informationn(s)=the number of ways an event can occur, assuming all ways are equally likely to occur. p(e) = n(e) n(s)
The following story, taken from the book by Polya, Patterns of Plausible Inference, Vol. II, Princeton Univ. Press, 1954, p.101, is also quoted in the book by Szekely, Classical paradoxes of probability
More information7.1 Chance Surprises, 7.2 Predicting the Future in an Uncertain World, 7.4 Down for the Count
7.1 Chance Surprises, 7.2 Predicting the Future in an Uncertain World, 7.4 Down for the Count Probability deals with predicting the outcome of future experiments in a quantitative way. The experiments
More information7.1 Experiments, Sample Spaces, and Events
7.1 Experiments, Sample Spaces, and Events An experiment is an activity that has observable results. Examples: Tossing a coin, rolling dice, picking marbles out of a jar, etc. The result of an experiment
More informationChapter 6: Probability and Simulation. The study of randomness
Chapter 6: Probability and Simulation The study of randomness Introduction Probability is the study of chance. 6.1 focuses on simulation since actual observations are often not feasible. When we produce
More informationApplications of Probability
Applications of Probability CK12 Kaitlyn Spong Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) To access a customizable version of this book, as well as other interactive
More informationCounting and Probability Math 2320
Counting and Probability Math 2320 For a finite set A, the number of elements of A is denoted by A. We have two important rules for counting. 1. Union rule: Let A and B be two finite sets. Then A B = A
More informationThe topic for the third and final major portion of the course is Probability. We will aim to make sense of statements such as the following:
CS 70 Discrete Mathematics for CS Spring 2006 Vazirani Lecture 17 Introduction to Probability The topic for the third and final major portion of the course is Probability. We will aim to make sense of
More informationGeneralized Permutations and The Multinomial Theorem
Generalized Permutations and The Multinomial Theorem 1 / 19 Overview The Binomial Theorem Generalized Permutations The Multinomial Theorem Circular and Ring Permutations 2 / 19 Outline The Binomial Theorem
More informationMATH 215 DISCRETE MATHEMATICS INSTRUCTOR: P. WENG
MATH DISCRETE MATHEMATICS INSTRUCTOR: P. WENG Counting and Probability Suggested Problems Basic Counting Skills, InclusionExclusion, and Complement. (a An office building contains 7 floors and has 7 offices
More information